Kinetic modeling of microscopic processes during electron cyclotron resonance microwave plasma-assisted molecular beam epitaxial growth of GaN/GaAs-based heterostructures

Microscopic growth processes associated with GaN/GaAs molecular beam epitaxy (MBE) are examined through the introduction of a first-order kinetic model. The model is applied to the electron cyclotron resonance microwave plasma-assisted MBE (ECR-MBE) growth of a set of delta-GaNyAs1–y/GaAs strained-layer superlattices that consist of nitrided GaAs monolayers separated by GaAs spacers, and that exhibit a strong decrease of y with increasing T over the range 540–580 °C. This y(T) dependence is quantitatively explained in terms of microscopic anion exchange, and thermally activated N surface-desorption and surface-segregation processes. N surface segregation is found to be significant during GaAs overgrowth of GaNyAs1–y layers at typical GaN ECR-MBE growth temperatures, with an estimated activation energy Es ~ 0.9 eV. The observed y(T) dependence is shown to result from a combination of N surface segregation/desorption processes

alphaCertified: certifying solutions to polynomial systems

Smale's alpha-theory uses estimates related to the convergence of Newton's method to give criteria implying that Newton iterations will converge quadratically to solutions to a square polynomial system. The program alphaCertified implements algorithms based on alpha-theory to certify solutions to polynomial systems using both exact rational arithmetic and arbitrary precision floating point arithmetic. It also implements an algorithm to certify whether a given point corresponds to a real solution to a real polynomial system, as well as algorithms to heuristically validate solutions to overdetermined systems. Examples are presented to demonstrate the algorithms.Comment: 21 page

Maximum Likelihood for Matrices with Rank Constraints

Maximum likelihood estimation is a fundamental optimization problem in statistics. We study this problem on manifolds of matrices with bounded rank. These represent mixtures of distributions of two independent discrete random variables. We determine the maximum likelihood degree for a range of determinantal varieties, and we apply numerical algebraic geometry to compute all critical points of their likelihood functions. This led to the discovery of maximum likelihood duality between matrices of complementary ranks, a result proved subsequently by Draisma and Rodriguez.Comment: 22 pages, 1 figur

On deflation and multiplicity structure

This paper presents two new constructions related to singular solutions of polynomial systems. The first is a new deflation method for an isolated singular root. This construction uses a single linear differential form defined from the Jacobian matrix of the input, and defines the deflated system by applying this differential form to the original system. The advantages of this new deflation is that it does not introduce new variables and the increase in the number of equations is linear in each iteration instead of the quadratic increase of previous methods. The second construction gives the coefficients of the so-called inverse system or dual basis, which defines the multiplicity structure at the singular root. We present a system of equations in the original variables plus a relatively small number of new variables that completely deflates the root in one step. We show that the isolated simple solutions of this new system correspond to roots of the original system with given multiplicity structure up to a given order. Both constructions are "exact" in that they permit one to treat all conjugate roots simultaneously and can be used in certification procedures for singular roots and their multiplicity structure with respect to an exact rational polynomial system.Comment: arXiv admin note: substantial text overlap with arXiv:1501.0508

A primal-dual formulation for certifiable computations in Schubert calculus

Formulating a Schubert problem as the solutions to a system of equations in either Pl\"ucker space or in the local coordinates of a Schubert cell typically involves more equations than variables. We present a novel primal-dual formulation of any Schubert problem on a Grassmannian or flag manifold as a system of bilinear equations with the same number of equations as variables. This formulation enables numerical computations in the Schubert calculus to be certified using algorithms based on Smale's \alpha-theory.Comment: 21 page